\subsection{Numerical method for finding fixed points}

The results reported in this section depend on the calculation of positive
steady states for networks with single terminal-linkage class. We now describe the algorithm
we use to solve for the associated fixed points.


Given an initial point $0<v_0\in \Re^n$, and a small tolerance $\tau$, the algorithm proceeds as follows:
\begin{algorithmic}
  \STATE $v^\star \gets v_0$
  \STATE $\mu \gets YA^Tv^\star$
  \WHILE{$\|YA_k v^\star\|_\infty>\tau$}
  \STATE $v^\star\gets \text{unique solution of \eqref{convex-fix}} $
	\STATE $\mu \gets \frac{1}{2}\mu +\frac{1}{2}YA^Tv^\star$
  \ENDWHILE
  \label{fixpoint-alg}.
\end{algorithmic}

The evaluation of the minimization step requires solving the linearly
constrained convex optimization problem \eqref{convex-fix}. Our implementation uses the PDCO
package \cite{pdco} for this purpose. 

Provided that the iterates satisfy $v^\star(\mu_k)>0$, and that
the minimization step is solved with sufficiently high accuracy, the optimality
conditions for \eqref{convex-fix} will imply that
\[\|Y^T\lambda-\log(v^\star(\mu_k))\|_\infty \leq \delta,\] for some small value of
$\delta$. Thus, if the iteration converges, at the fixed point \eqref{mak} will
be satisfied to a precision $\delta$ and \eqref{fb} to a precision $\tau$.

The aforementioned algorithm has been tested extensively on randomly generated
networks, with noteworthy success. Section \ref{scn:convergence} reports our
experiments.
